In the study, we have introduced two anomaly free gauge symmetries i.e. $U(1)_{B-L}$ and $U(1)_{L_e-L_\mu}$ in an inverse seesaw framework as extension of Standard Model. Inclusion of three right handed neutrinos $N_{R_i}$ and three left handed fermions $S_{L_i}$ explains neutrino phenomenology, neutrinoless double beta decay, electron and muon anomalous magnetic moment successfully. In addition to six new fermions, we have taken two scalar singlets $\chi_1$ and $\chi_2$ which break two extended gauge symmetries respectively. Two gauge bosons $Z_1, Z_2$ are used to explain $(g-2)_{ e, \mu}$ where $m_{Z_1}$ is in $\rm MeV$ range and $m_{Z_2}$ is in $\rm GeV$ range. Idea for taking such type of gauge bosons mass ranges is verified by the upper limits of so many current experiments like Orsay, E141, E774, NA64, KLOE, Babar inv and future experiments like FASER, VEPP3, DUNE etc.